The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

IF 2.2 2区数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Mathematical Programming Pub Date : 2024-03-13 DOI:10.1007/s10107-024-02069-0

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Abstract

In this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

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整数 SDP 的 Chvátal-Gomory 程序及其在组合优化中的应用

摘要本文研究了著名的整数半定式程序（ISDP）的 Chvátal-Gomory (CG) 过程。我们证明了通过迭代该程序所得到的松弛层次的几个结果。我们还研究了谱的基本封闭的不同形式。通过利用 SDP 的总对偶积分性，我们得出了特定类型谱的基本封闭的多面体描述。此外，我们还展示了如何在分支切割框架中利用（加强的）CG 切割来处理 ISDP。与文献中的现有算法不同，我们方法中的分离例程同时利用了半有限性和积分性约束。我们为组合优化问题中常见的几类二元 SDP 提供了分离例程。在论文的第二部分，我们介绍了我们的方法在二次旅行推销员问题（QTSP）中的综合应用。基于有向哈密顿循环的代数连接性，我们引入了两个模拟 QTSP 的 ISDP。我们证明，由这些公式产生的 CG 切分包含几个著名的切分平面族。数值结果表明了 CG 切分在我们的分支-切分算法中的实用性，该算法的性能优于其他 ISDP 求解器，能够最优地求解大型 QTSP 实例。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Programming 数学-计算机：软件工程

CiteScore

5.70

自引率

11.10%

发文量

160

审稿时长

4-8 weeks

期刊介绍： Mathematical Programming publishes original articles dealing with every aspect of mathematical optimization; that is, everything of direct or indirect use concerning the problem of optimizing a function of many variables, often subject to a set of constraints. This involves theoretical and computational issues as well as application studies. Included, along with the standard topics of linear, nonlinear, integer, conic, stochastic and combinatorial optimization, are techniques for formulating and applying mathematical programming models, convex, nonsmooth and variational analysis, the theory of polyhedra, variational inequalities, and control and game theory viewed from the perspective of mathematical programming.